feat(agentStudio): surface thread depth error

[pipeline1987]

🤔 What

Improves Ask AI in @algolia/sitesearch:

• new sidepanel experience,
• Agent Studio / thread-depth handling, and chat + markdown polish.
• Docs registry hooks and the vanilla demo are updated to match.

How 🧐

• Ask AI core: thread-depth / Agent Studio–related behavior and messaging (see askai.ts, error-utils.tsx, index.tsx, search-input.tsx).
• Sidepanel Ask AI: new entry (sidepanel-askai), portal + floating panel shell, footer aligned with docsearch-style AI disclaimer and Powered by Algolia.
• ChatWidget: optional showAiDisclaimer / newestExchangeFirst; chronological order for sidepanel with stick-to-bottom scroll when appropriate.
• Markdown / code: theme tokens for dark-mode code blocks and inline code; horizontal overflow fixes (flex min-width: 0, constrained snippets, pre scroll).
• Vanilla: sidepanel-askai init, shared resolve-container-element helper; search/search-askai init aligned.

🧪 How to test

1. On Ask AI, hit thread-depth limit → see error + start new conversation; confirm the next turn uses a new API thread, not the capped one.
2. In the modal, send several follow-ups (Enter) → same conversation until you explicitly start new or leave Ask AI.
3. Sidepanel: open vanilla demo (or your integration), dark + light, long code blocks, new vs old messages order.

📸 Screenshots

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[codacy-production]

Up to standards ✅

🟢 Issues 0 issues

Results:
0 new issues

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🟢 Metrics 47 complexity · 6 duplication

Metric	Results
Complexity	47
Duplication	6

View in Codacy

_{TIP This summary will be updated as you push new changes. Give us feedback}

[LorrisSaintGenez]

A few comments on the experience:

• We should try to match the pre-existing React design

React	VanillaJS

• When clicking the LeftArrow on thread error, we are "blocked" until we exit the modal entirely to create a new convo. Clicking the arrow should allow us to start a new convo directly

Screen.Recording.2026-04-10.at.12.10.01.mov

• Not sure if we can get something about it, but it's a bit frustrating when the message we wrote is not visible even when we hit the thread depth limit. We should display it anyway like we do on AgentStudio

Screen.Recording.2026-04-10.at.12.10.30.mov